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Igumnov L. A., Metrikin V. S. On the Complex Dynamics in Simplest Vibrational Systems with Hereditary-Type Friction. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 4, pp. 433-446. DOI:


On the Complex Dynamics in Simplest Vibrational Systems with Hereditary-Type Friction


The dynamics of a number of vibrational systems, accounting for the forces of hereditary-type dry friction and a vibration limiter, are studied in the paper. The interaction between the vibration limiter and the vibrational system is assumed to obey Newton's hypothesis. A general mathematical model has been developed, which is a strongly nonlinear non-autonomous system with a variable structure. The dynamics of the mathematical model is studied numerically-analytically, using the mathematical apparatus of the point mapping method. The special feature of the studying approach is that a point map is not formed in a classical way (mapping Poincare surface into itself), but based on times of the relative rest of the vibrational system, which considerably simplified both the point mapping process and its detailed analysis. The presence of floating  boundaries  of plates of sliding motion required an original approach to point mapping and interpreting the results obtained. The developed investigation methodology and software product were used to study the phase-plane portrait of the mathematical model as a function of the characteristics of sliding friction forces and rest, as well as of the type and position of the limiter. Based on the character of the bifurcation diagrams variation, it was possible to find the main laws of the motion regimes alteration process (the occurrence of periodic motion regimes of arbitrary complexity and possible transition to chaos via the period-doubling process) with the changing parameters of the vibrational system (the amplitude and frequency of the periodic effect, forms of the functional relation describing the variation of the friction coefficient value of relative rest. The results obtained with and without accounting for a vibration limiter are also compared in the paper.


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